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I have done the general and specific terms as shown below.

Im not sure if these are correct?

thanks

General Terms

cos x = ∑ (-1)n (x^(2n) / (2n)!)

COS x = ∑ (-1)n (x^(2n+1)/ (2n +1) / (2n)!)

sin x = ∑ (-1)n (x^(2n+1) / (2n+1)!)

‘sin x = ∑ (-1)n (((2n+1) x^(2n))/ (2n+1)!)

Specific Terms

sin x = x – (x^3 / 3!) + (x^5 / 5!) – (x^7 / 7!) + (x^9 / 9!) - …

‘sin x = 1 – (3x^2 / 3!) + (5x^4 / 5!) - (7x^6 / 7!) + (9x^8 / 9!) - …

cos x = 1 – (x^2 / 2!) + (x^4 / 4!) – (x^6 / 6!) + (x^8 / 8!) - …

COS x = x – (x^3 / 3 / 2!) + (x^5 / 5 / 4!) – (x^7 / 7 / 6!) + (x^9 / 9 / 8!) - …

thanks